Single-shot and measurement-based quantum error correction via fault complexes

TL;DR

The paper introduces fault complexes as a homological, dynamic framework for describing foliation-based fault-tolerant graph states in measurement-based quantum computation. By tensoring a base CSS code with a repetition code, it derives explicit formulas for primal/dual correlations, errors, and fault distances, enabling precise analysis of single-shot decoding and decoding improvements. Numerical results show elevated thresholds for 3D and 4D toric codes under phenomenological and photonic GKP noise, with overlapping-window decoders achieving strong performance and stability experiments illustrating practical lattice-surgery advantages. The framework also extends to subsystem codes and circuit-model noise, and even demonstrates foliations beyond the repetition code, highlighting potential for higher-dimensional codes and photonic implementations to realize scalable, fault-tolerant MBQC architectures.

Abstract

Photonics provides a viable path to a scalable fault-tolerant quantum computer. The natural framework for this platform is measurement-based quantum computation, where fault-tolerant graph states supersede traditional quantum error-correcting codes. However, the existing formalism for foliation - the construction of fault-tolerant graph states - does not reveal how certain properties, such as single-shot error correction, manifest in the measurement-based setting. We introduce the fault complex, a representation of dynamic quantum error correction protocols particularly well-suited to describe foliation. Our approach enables precise computation of fault tolerance properties of foliated codes and provides insights into circuit-based quantum computation. Analyzing the fault complex leads to improved thresholds for three- and four-dimensional toric codes, a generalization of stability experiments, and the existence of single-shot lattice surgery with higher-dimensional topological codes.

Figures (6)

• Figure 1: Foliation of the surface code viewed as a fault complex. (a) The distance-3 surface code Tanner graph with circles representing qubits, squares representing $X$ and $Z$ checks, and dashed and solid lines representing the connectivity of these checks, respectively. (b) and (c) Hypergraph product of (a) with a repetition code check and bit node, respectively. This yields two types of fault locations, teal and yellow, along with primal (red) and dual (purple) detectors. Dashed and solid lines indicate boundary maps of the fault complex. For simplicity, we omit out-of-plane connections. (d) Unit cell of the fault complex obtained by stacking alternating layers of (b) and (c). A dual detector (purple) is formed by the parity of six dual fault locations (yellow). Vertical red dashed lines are omitted for clarity. (e) Mathematical structure of the fault complex and its relation to (b)-(d): colored circles and squares represent chain complex vector spaces and lines represent boundary maps.
• Figure 2: Sustainable thresholds for 3D (a) and 4D (b) toric codes under phenomenological Pauli noise. Markers represent the threshold for a $(w, 1)$-overlapping window decoder as a function of noisy syndrome rounds. The black dashed line shows the threshold for the optimal window choice $w = L$; see supp for details.
• Figure S1: Sustainable thresholds for 3D (a) and 4D (b) toric codes under a noise model for a photonic GKP-based architecture. Markers represent the threshold for a $(w, 1)$-overlapping window decoder as a function of noisy syndrome rounds. The black dashed line shows the threshold for the optimal window choice $w = L$. In the 3D case, we only consider primal faults, whereas in the 4D case, we consider both primal and dual faults.
• Figure S2: Example of the threshold fitting procedure for the 3D toric code under the photonic GKP noise model with an $w=2$ overlapping window decoder after $129$ rounds of noisy syndrome measurements. Left: Histogram of the bootstrap resamples, with the mean and $99\%$ confidence interval highlighted in blue. Center: Collapse plot of the fitted data. Right: Threshold plot of the data obtained from the sustainable threshold simulation with the threshold value highlighted as a black dashed line and $99\%$ confidence intervals as a grey shading.
• Figure S3: Logical error rate for the time-like logical correlation of a 3D toric code for different window sizes of the overlapping window decoder. In (a)$w = 1$, (b)$w = 2$, and (c)$w = 3$. Vertical black dashed lines indicate the fitted threshold. The shading indicates a confidence interval, indicating hypotheses whose likelihoods are within a factor of 1000 of the maximum likelihood estimate.